John, nowadays logging into your bank on someone else's PC would be considered extremely risky! Better to do a bank transfer on your own mobile or laptop, and don't connect to their wifi either.
Please explain what the extreme risk is, especially when nowadays there is an additional security hurdle of a text message to my phone to access the account. Even if the vendor was so clever as to extract the keystrokes of the first hurdle (d.o.b. plus secret number and the 4 digit pin, numbers of which are entered in varying order), his computer could not possibly divine the password from three random characters. And for a strange payee a final hurdle in the form of a card reader and bank card (another PIN needed) are required.
Because if you are accessing your account via a third party's computer, they can have either their own or be affected by (for example) a 'keylogger' virus or similar program (never mind the owner actually seeing/recording what characters you type for the password/codes) and thus will then have full access to your account.
It's the same method used by fraudsters who use card cloning devices (with a video camera on the side to get the PIN) on ATMs to get access to people's accounts without having to physically steal their bank or credit card.
To pay for a car, best to get all the paperwork done first (including them prove who they are and that the address they are selling from is genuinely their home), and then get the transfer done at the bank/building society.
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